An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that a′(G)≤ +2, where = (G) denotes the maximum degree of the graph. Alon e...

A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ ∆(G) + 2 where ∆(G) is the maximum degree in G. It is known that a′(G) ≤ 16∆(G) for any graph G (see [2],[10]). We prove that there exists a const...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph is called 2-degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2-degenerate graphs properly contain...

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An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph G is called fully subdivided if it is obtained from another graph H by replacing every edge by a path of length at least two. Fully subdi...

The r-acyclic edge chromatic number of a graph is defined to be the minimum number of colours required to produce an edge colouring of the graph such that adjacent edges receive different colours and every cycle C has at least min(|C|, r) colours. We show that (r − 2)d is asymptotically almost surely (a.a.s.) an upper bound on the r-acyclic edge chromatic number of a random d-regular graph, for...

We provide an alternative constructive proof of the Asymmetric Lovász Local Lemma. Our proof uses the classic algorithmic framework of Moser and the analysis introduced by Giotis, Kirousis, Psaromiligkos, and Thilikos in " On the algorithmic Lovász Local Lemma and acyclic edge coloring " , combined with the work of Bender and Richmond on the multivariable Lagrange Inversion formula.